Humanity has recently passed another amazing milestone: inexpensive phone cameras are now better than the human eye. In other words, most of us are now walking around with a tool that can augment our vision.

For me, this happened with the new phone I bought about 10 months ago. Slowly over those months, I realized the power of the camera. The phone is a middle-of-the-line Samsung.

It helps that I've been taking pictures, amateurishly, for about 15 years now. I've been squeezing my old Sony Cybershot for every bit of performance it could give me. So when I got a new camera, I knew how to push its limits.

It also helps that my work sometimes involves taking pictures of fine details.

With all that said, here are some of my photos from the last few months.

Neurons! (As seen through a glass bead.)

This is Hark. He's our local herald angel. Any guesses as to how we got this shot?

This mysterious photo of the Costanera tower was not edited. This is exactly how it looked.

On the left is my phone, with the help of a lamp from our living room. On the right is a professional camera taking a photo of a penny inside a "light box" -- supposed to be great for bringing out the detail for small object photos.

Bottom line: the magic is probably already in your hands. Go out and take some great pictures!

Doodle told me this morning that he has been "really enjoying triangles", but he had some inquietude about the Law of Cosines ... he couldn't quite get why and how it works.

It is one thing to be able to use a formula, yet another to prove or derive it. But often far beyond both of those is the ability to internalize it -- to have a mental map of its operation.

We hacked around with it for a while, trying to get some traction on it. After about 30 minutes, we both felt satisfied that we had our heads around it.

It struck me that he is habituated to understanding everything he learns, and it is always a let-down to him if we have to just accept something, lacking the ability to see inside some "black box" of knowledge. I could hardly have formed such a habit in a school setting, because the proofs and formulas came on schedule -- too fast to internalize -- and no one prepared me for such a task.

I never even internalized the Pythagorean Theorem, yet supposedly I was one of the top math students in my state.

(Can you use the images below to do this? See here and here for more.)

"Other folks trying to understand the Law of Cosines would really benefit from a little visualization app, like you did for the eclipse shadow," I said.

He agreed and seemed to think it wouldn't be too hard. Then we pondered about how it would be for him to do a family of apps devoted to demonstrating complicated concepts.

With limited help from the local web goddess (Sudoku), he recently started into the long slog of setting up for coding Android apps. Many headaches and investigations later, he can make a smiley face app and send it to my phone.

Yet, I have no doubt that he will soon be blessing the app store with some little treasures. He has a calm persistence and a real love of creating these works. And he's eager to be working in a serious coding environment where he has good tools at hand. (unlike his current situation, pouring his work into the beginner-oriented Khan Academy platform)

What is the common thread of all of the above? It is that he is blooming in a low-structure "pull environment" where he determines his pace and has room for chasing his curiosities.

So, being Thanksgiving 2017, I declare that I am thankful to the modern pioneers of homeschooling (parents from the 60's and 70's who took big risks) who helped make our wild, wonderful unschooling world possible.

[Rosebud & Sudoku] Chile's independence day was Monday, September 18, and Sunday (September 17), was national Cueca day, Chile's national dance. Each year there are traditional foods and activities at our local parks. In past years, there has been a huge celebration at our closest park, lasting several days with lots of people and LOUD music into the night.

This year, our local event was more quiet and less crowded, which was very nice.

We've always enjoyed watching the Cueca, but we hadn't learned how to do it before this year. Dad helped us get started learning (while teaching himself!) The timing is the trickiest part, but we slowly got the hang of it. Once we had practiced at home, we headed over to the park.

This is a picture , which was posted on Twitter by the Providencia mayor, Evelyn Matthei, of the Cueca area. We think this is her mother dancing the cueca like a pro. Evelyn was the runner-up in the 2014 presidential election here in Chile. If you look closely, Carman and I are in the background. (I'm in red jeans, and Carman is wearing a plaid shirt and a hat.)

The Cueca class we did was very helpful. The teacher started everyone out in a large circle and walked us through the steps. Then we all did the steps in the circle by ourselves (instead of with a dance partner). The teacher counted out the steps and told us when to switch. Here are the steps and count that goes with them:

After going through the steps a few times in the circle, the teacher told us to split into pairs. I (Sudoku) got to dance with a guy all dressed up in the typical Cueca attire (He saw that I was REALLY inexperienced, and helped me out).

[By Doodle] There were many traditional dieciochera games.

There's that one where you smash a lever with a big hammer and it throws up a little bobbin that zooms upward on a cable and you try to get it to touch the bell at the top. Dad pointed out that it's mostly luck. It's mostly about how much wobble is put on the cable. The less wobble, the more freely the bobbin can zoom up.

We watched some big, buff dudes do it for a little bit, but none of them could get it past 4 and a half. (The bobbin starts at zero, and the bell is at five.)

Then, there's a game where some glass bottles are put in some holes and you have to throw some wooden rings around the tops of the bottles that are sicking out through the holes.

There's a smash the kittens game. Three years ago, they had a reward. There are five stuffed kitties and four balls. If you smashed four of the kitties off the shelves with your four balls, you got a sandwich with a piece of laminated cheese, a piece of meat (the really thin kind), and two pieces of "rock bread" as we call it to hold the cheese and the meat together.
I actually won that prize!

The kitties are pretty big and light, but the game is pretty hard, because the balls are so light and you have to throw the balls from pretty far away. I got a kitty with my first and second shot, missed with my third shot, and got two at once on my fourth shot.

Yeah. I'm sure it was complete luck.

Normally they have tug of war and a game of unwinding ropes, but they didn't have that this year, either.

Both this year and the last year, Sudoku and I went to go do the bowling game. Normally there are park employees there to set up the pins for people, but there wasn't anybody there at the time. The two people who were supposed to be doing the pin-setting-up were lounging a few yards off to the side and had their faces buried in their phones.

We went up and set up the pins. Before we could bowl any, some people came over, also wanting to bowl.
They were first in line, so we let them go first.

But, before the people were done bowling, and before Sudoku and I were done setting up the pins for them, even more people got in line.

After a while, we were the ones managing the bowling place. And, for some STRANGE reason, *cough, cough* the bowling place suddenly got a whole lot more popular.

Tonight, just before supper time, 10 year-old Rosebud pulled up to the piano to play the song she had been working on today. It was "Yellow Ribbon".

After playing it, she announced the title of the piece and both Sudoku (17) and Doodle (12) got quizzical looks on their face.

"But Dad, that's not the way Zeb sang it!" they each said in their own way.

I was staring back at them like they were crazy.

It dawned on me that, in their memory, they had actually heard the big, calm cowboy Zeb, riding on his small horse, sing that tribute to Pike's Peak as the herds slowly wended their way through the ravines, in sight of the summit.

...when they were riding along with him back in about 1910.

Wow.

Though I've probably read only 1% as many books to the kids as their mom has, I make up for it with my energy and craziness.

When I come to a song in a book, I sing it! If I don't know the tune, I make one! I march along with soldiers, do one-person re-enactments of bar fights, throw things, bite myself ... okay, well, I've never actually done that last one, pretty sure. But you get the idea.

Anyway, when we were reading this a few years back, I did the best I could to sing the Yellow Ribbon tune, but I missed it by ... a fur piece.

Nonetheless, I sang with conviction, and I imitated the drawl that Zeb was said to have. And, since that song appears many times in the text, my kids became quite familiar with my version.

And, magically, the kids heard Zeb himself singing through me. Like they were going back in time.

As Hank, Zeb's crotchety old compadre would say in his geezerly whine, "Back in my day, ..."

[thanks to Flickr's brando.n for the shot with Pike's Peak in the distance]

This weekend in Providencia we had snow and expectations of a hard freeze on the following night. We were closely watching the forecast for that night from as long as a week beforehand. At one point we saw a prediction of -9°C. I'm pretty sure that would have broken the record low temperature for Santiago.

As we got closer to the day, the forecasts moderated somewhat, and, the evening before the expected freeze, forecasts were saying -3 or negative 4 degrees Celsius. That's still quite cold for here -- probably would be the coldest temp of the winter.

We have an exposed laundry room (known here as a logia) that has a head-high concrete wall around it on the two exterior sides -- from there up to the ceiling it is open. The wall is 6 or 7 inches thick. The plumbing for our washing machine and an adjoining sink runs inside of that wall, so it was going to have the outdoor temperature on both sides of the wall/plumbing. I was pretty nervous about how to handle the situation. I could run a trickle of water from all those taps, but some of them like to "turn themselves off" when you walk away.

I was also concerned about needing to completely drain the washing machine and the hot water heater, and I would need to think about the vegetables and reserve water that we store in that area.

Doing this planning took my mind back many years to memories of a hard freeze we had in Chattanooga when I was a child.

We had just moved south. All of my family history comes from the great plains, the northern Midwest, and Canada, so my dad knew plenty about how to handle deep freezes. But it turned out that the waterline to our house was not buried deep enough. I suppose if we had been running a trickle of water inside the house (for all I know we may have been), that might have been enough to keep that main water line from freezing, but clearly that was not his fault.

I remember the hardships of digging and hacking at the frozen soil to pull up the burst pipe and then to bury another one deeper. I was too young to actually be a help, but I was too old not to be out there trying to help. Tempers were short. Icy mud was aplenty. There was a sense of urgency, for some reason, as I recall.

In thinking of this, I just realized it was one of the only memories I have of seeing my brother work or working with him. (I did see him play the piano for my Dad's evangelistic meetings, but that's not the kind of work I mean.) He would have been 13 or 14 at the time -- a lot older than me.

The memory instilled in me a healthy fear of freezing pipes.

My other big burst-pipe memory comes from about a year after Milkmaid and I were married. We were living in a basement apartment and we were about to walk out of the apartment and head to a family holiday gathering when suddenly water began to pour down through the ceiling into our apartment.

It turns out that the renters above us had made the mistake of closing their laundry room door. The consequence of this was that the warm air from the house was not able to keep the plumbing lines from freezing.

It seems like a very bad time to have such a thing happen, but of course there is never a good time for such things.

So, back to yesterday in Chile, I decided to do nothing and simply check on things as the night progressed. I felt that -4°C might not be enough to freeze anything in just one night.

During the night, I made a happy discovery. I realized that the “heat envelope” around our large apartment building was very significant, and our laundry room was not going to go below freezing.

Here is my rule of thumb going forward:

On the higher floors of a large residential building here in Santiago (this particular one built in 2000), it seems that you get a thermal bonus of about one half ° Celsius or 1°F for every floor below you. So if you were on the 11th floor, the temperature outside your windows on a cold night could be expected to be about 5°C above the street temperature.

These numbers would not work for a well-insulated building, but, because Santiago typically only drops below freezing a few times a winter, the insulation is poor and a lot of heat is lost from the windows (and walls?) on the lower floors.

I knew there must be some effect like this, but I had no idea it was so large. There must be great plumes of heat rising from all the large buildings on a cold night.

So for anyone thinking of living in Santiago, I recommend that you prefer the higher floors if you are on a tight budget like us. Locals tend to dislike the higher floors because they get more earthquake action.

There was a interesting article in the newspaper, and the headline was: "If there were 100 people in Chile." Inside the article were percentages of how many people did this, and how many people did that...

So here are the most interesting things about it.

51 are women and 49 are men

19 identify themselves with a political party

76 say they have pardoned those who have done them wrong

68 have access to mobile internet

6 are unemployed

58 say they are Catholic

18 say they are evangelicals

20 say they are atheists

3 say they have other beliefs

12 run for exercise

20 rent their house

64 own their house

3 say they have bad health

34 voted in the last election

93 don't trust the political parties

13 live in rural areas

8 work in agriculture

59 consider themselves happy

12 consider themselves sad

20 have some disability

4 consider themselves Mapuche (but the guy who is in the picture has blue eyes!)

We were talking about solar eclipse phenomena recently. I told the kids about my memories from eclipses that I experienced as a boy, especially about how all the leaves of a tree would take on the shape of the eclipse.

After the conversation, I did some image searches so I could show them what I remembered.

I was stunned to find that the truth was nothing like I remembered it!

Instead, there are thousands of images showing pinhole-camera effects. That is, the light from the sun passes through tight squeezes between the leaves and casts itself on the ground as an inversion of the sun's distorted shape. Here is a typical example (Photo by torbakhopper):

I couldn't find a single picture to support what I remembered!

But I couldn't seem to let go of it either. Without being able to explain very well why, it just seemed to me that there would be an eclipse shadow effect separate from the well-known pinhole effect.

I muttered something to the effect of, "We'd need some fancy lighting or a computer program to know if there was any truth to what I was thinking."

Little did I know that wheels had been set in motion...

Over the past week, I have been working on a project. A coding project.

Sometimes we would play games with light during supper, when the light from the sun would reflect off glass buildings. I noticed that the shadows were almost perfectly crisp, even though our shadows fell on a wall that was 15-20 feet away.

That isn't the case when the light is coming straight from the sun.

In the morning's direct sunlight, the shadows are all fuzzy, and they would do all kinds of crazy stuff, like jumping over to other shadows, or some shadow that is a lot thinner than it should be...

A week ago, I had a realization about why shadows seem to warp sometimes.

I had always assumed that it was from the light of the sun bending, slightly.

But that isn't the case. I realized that if you have a small slit letting sunlight through, there will be a light spot on the ground that is a good bit wider than the small slit. I realized this was because of the light from the right side of the sun shining through to the left of the slit, and vice versa.

During an eclipse, also, a tree shows many mini eclipses on the ground, on it's shadow. This, we found out, is due to the pinhole effect.

Marathon still felt that a normal object, without a pinhole might also give an eclipse-like shadow. We made this big sketch, we kept messing up, but finally, we were pretty sure that any object would give a slightly eclipse-like shape.

It would take a long time to explain it all in writing. It's pretty complicated.

So, instead, I did this coding project to make it easier to understand:

I consider ~150 inches from the ground to be the best distance for seeing the eclipse-like shadow.

Hope you like it!

[Marathon: So, thanks to Doodle's javascript program, we can see how a shadow of basically anything leaf-sized, that is positioned around 12 feet above the ground, gets distorted by parallax effects to look vaguely like the crescent of the eclipse.

Here's a leaf-like shape's shadow at 10 inches above the ground during an ~80% solar eclipse:

Now here's the same shape's shadow at the same moment if it was 12 feet off the ground:

Totally different!

Even the staple shape, that already is a crescent of sorts, will bend to roughly become a crescent in the opposite direction! Try it for yourself.

Here's another neat effect we found. If you set the moon's size to be slightly smaller than the sun so that it allows for a "ring of fire" at the point of complete eclipse, here's what the shadow of a ping-pong sized ball looks like.

The locals use the word "Andinismo" to mean "exploring the Andes Mountains". So far our experience of the Andes has been limited to areas in close proximity to civilization.

I felt it was time to do something more, and summer was almost gone.

Following a two day scouting trip by Doodle and I a few weeks ago, Sudoku joined us this past Sunday for a three day attempt to reach the La Paloma glacier, the lesser of the two glaciers visible from downtown Santiago.

streams through needle-like tundra grass

We picked the warmest sequence of days we could find, but the temperature still dropped below freezing at night where we slept.

The 20 km trail toward the glacier was quite busy on Sunday afternoon, but once night came at our base camp around 9000 feet, it would not be until midday Tuesday that we would see another human being.

It's safe to drink the water at certain places along the hike. Here's a little log of that for those interested:

At 1.5 hours of ascent, you'll cross the potable Agua Larga, marked with a wooden sign.

At four hours, you'll reach La Lata, a marshy pasture area with horses. The water here is said to be safe to drink, but I didn't trust it because of the presence of horse and hiker feces in the area. However, just above La Lata, there is a stream much like Agua Larga which shows none of the copper/ sulfur discoloration indicative of the non-potable waterways. We took water at that stream.

At six hours, you'll reach Las Cascadas, where two major waterways join, but none of the water in that area is considered potable due to the high concentration of metals.

At seven hours, Piedra Carvajal appears to be a former glacier lake, but now filled with tundra and very clear streams of water running through it. This water is said to be safe, and it's your last chance for water before the glacier. [Sudoku: The grass here was so tough that some spots of blood appeared on my hand when I touched it!]

The hike was hot and dusty and we had some boot problems and other difficulties with our cobbled-together gear. Fortunately, the three of us are all fairly close on shoe size right now, so we were able to swap around to mitigate the effects of boot irritations.

[Sudoku: My toes only came within about 2.5" of the end of Dad's boots, but they were pretty comfortable.]

By the time we reached our intended overnight spot, La Lata, the air had grown noticeably thin.

The peaks ahead of us were amazing under light of sunset. The stars would have been great (Doodle saw five shooting stars on our scouting trip), but we had a dusk that seemed to last forever and a full moon.

Here in Chile, we've grown quite accustomed to having grazing animals around us when we slept outside in the countryside. Nonetheless, it was a bit unnerving to have horses nonchalantly grazing right up against our tent all through the night.

[Sudoku: I woke up several times to hear Dad shooing the horses away from our tent again.]

We woke feeling good on Monday morning and went for the glacier. It was clear and dry. (there was no dew on the tent.) I couldn't keep up with the kids, so I stopped about an hour short of the glacier's scree-field base and they went to the glacier without me. (And I had the camera with me. Hence no actual pictures of Paloma glacier. Sorry! Here are some.)

camping with grazing animals

On the way down we had the thrill of watching a pair of soaring/gliding condors -- practically stationary in the air just above us -- as we approached one of the steepest sections of the trail.

I never saw a mouse, but I did see a hole in a bag of cheese left inside a backpack we hid in the rocks back at the camp site.

the enchanted tundra field at Casa Piedra Carvajal

Here are some of my takeaways regarding Andinismo:

we form a route by connecting water points like camels in the desert and those water points are often glorious

the tremendous scale of everything boggles the mind, and the lack of vegetation makes it very difficult to tell how far way things are because there is no reference object

managing the sun and the cold is difficult but doable, even for lightly-equipped folk like us

[Sudoku: The boys did a great job making sure we had everything we needed for the trip. They have short hair though, and weren't able to warn me about the big tangle I had by the end of our trip!]

[Doodle: Once, when we were hanging out around some small pools of red water, I noticed that it wasn't the water that was red, but that there were hundreds of tiny, red bugs, all crawling over each other.]